Elementary Number Theory: Primes, Congruences, and Secrets

A Computational Approach

Author: William Stein

Publisher: Springer Science & Business Media

ISBN: 0387855254

Category: Mathematics

Page: 168

View: 6265

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem.

Elementary Number Theory

An Algebraic Approach

Author: Ethan D. Bolker

Publisher: Courier Corporation

ISBN: 0486153096

Category: Mathematics

Page: 208

View: 6807

This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and more.

Elementary Number Theory

Second Edition

Author: Underwood Dudley

Publisher: Courier Corporation

ISBN: 0486134873

Category: Mathematics

Page: 272

View: 4686

Written in a lively, engaging style by the author of popular mathematics books, this volume features nearly 1,000 imaginative exercises and problems. Some solutions included. 1978 edition.

Elementary Number Theory

Author: Gareth A. Jones,Josephine M. Jones

Publisher: Springer Science & Business Media

ISBN: 144710613X

Category: Mathematics

Page: 302

View: 527

An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.

Elementary Number Theory with Applications

Author: Thomas Koshy

Publisher: Elsevier

ISBN: 9780080547091

Category: Mathematics

Page: 800

View: 9173

This second edition updates the well-regarded 2001 publication with new short sections on topics like Catalan numbers and their relationship to Pascal's triangle and Mersenne numbers, Pollard rho factorization method, Hoggatt-Hensell identity. Koshy has added a new chapter on continued fractions. The unique features of the first edition like news of recent discoveries, biographical sketches of mathematicians, and applications--like the use of congruence in scheduling of a round-robin tournament--are being refreshed with current information. More challenging exercises are included both in the textbook and in the instructor's manual. Elementary Number Theory with Applications 2e is ideally suited for undergraduate students and is especially appropriate for prospective and in-service math teachers at the high school and middle school levels. * Loaded with pedagogical features including fully worked examples, graded exercises, chapter summaries, and computer exercises * Covers crucial applications of theory like computer security, ISBNs, ZIP codes, and UPC bar codes * Biographical sketches lay out the history of mathematics, emphasizing its roots in India and the Middle East

Elementary Number Theory

Author: N.A

Publisher: McGraw-Hill Higher Education

ISBN: 0077418123


Page: N.A

View: 3168

Elementary Number Theory in Nine Chapters

Author: James J. Tattersall

Publisher: Cambridge University Press

ISBN: 9780521850148

Category: Mathematics

Page: 430

View: 9693

This textbook is intended to serve as a one-semester introductory course in number theory and in this second edition it has been revised throughout and many new exercises have been added. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject.

Elementary Number Theory with Programming

Author: Marty Lewinter,Jeanine Meyer

Publisher: John Wiley & Sons

ISBN: 1119062764

Category: Mathematics

Page: 232

View: 8245

Filling a much-needed gap in the current literature, this book expertly bridges the subjects of number theory and programming and features a multitude of examples and programming exercises in each chapter. It provides an introduction to elementary number theory with fundamental coverage of computer programming and is appropriate for students of mathematics and computer science alike who need to become acquainted with the most famous theorems, problems, and concepts of number theory. In addition, the authors provide a comprehensive presentation of the methodology and applications for readers with various levels of experience, and while theorems are provided, the authors avoid the standard theorem/proof format to aid in reader comprehension. The book features sample programs and research challenges at the end of each chapter for readers to work through, as well as an appendix that provides select answers to the chapter exercises. The authors also maintain a supplementary material website that provides additional working examples of the computer programs. Topical coverage includes: special numbers; Fibonacci sequence, primes, and the Pell equation; Pascal's triangle; divisors and prime decomposition; modular arithmetic; number theoretic functions; Euler's Phi function; sums and partitions; and cryptography. Prerequisites include basic algebra and some knowledge of any computer language.

Elementary Number Theory, Cryptography and Codes

Author: M. Welleda Baldoni,Ciro Ciliberto,G.M. Piacentini Cattaneo

Publisher: Springer Science & Business Media

ISBN: 9783540692003

Category: Mathematics

Page: 522

View: 488

In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter. Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.

Analytic and Elementary Number Theory

A Tribute to Mathematical Legend Paul Erdos

Author: Krishnaswami Alladi,P.D.T.A. Elliott,A. Granville,G. Tenenbaum

Publisher: Springer

ISBN: 1475745079

Category: Mathematics

Page: 300

View: 9152

This volume contains a collection of papers in Analytic and Elementary Number Theory in memory of Professor Paul Erdös, one of the greatest mathematicians of this century. Written by many leading researchers, the papers deal with the most recent advances in a wide variety of topics, including arithmetical functions, prime numbers, the Riemann zeta function, probabilistic number theory, properties of integer sequences, modular forms, partitions, and q-series. Audience: Researchers and students of number theory, analysis, combinatorics and modular forms will find this volume to be stimulating.

Elementary Number Theory

Author: James S. Kraft,Lawrence C. Washington

Publisher: CRC Press

ISBN: 1498702694

Category: Mathematics

Page: 411

View: 756

Elementary Number Theory takes an accessible approach to teaching students about the role of number theory in pure mathematics and its important applications to cryptography and other areas. The first chapter of the book explains how to do proofs and includes a brief discussion of lemmas, propositions, theorems, and corollaries. The core of the text covers linear Diophantine equations; unique factorization; congruences; Fermat’s, Euler’s, and Wilson’s theorems; order and primitive roots; and quadratic reciprocity. The authors also discuss numerous cryptographic topics, such as RSA and discrete logarithms, along with recent developments. The book offers many pedagogical features. The "check your understanding" problems scattered throughout the chapters assess whether students have learned essential information. At the end of every chapter, exercises reinforce an understanding of the material. Other exercises introduce new and interesting ideas while computer exercises reflect the kinds of explorations that number theorists often carry out in their research.

Elementary Number Theory

Author: Charles Vanden Eynden

Publisher: Waveland PressInc

ISBN: 9781577664451

Category: Business & Economics

Page: 278

View: 9734

A Guide to Elementary Number Theory

Author: Underwood Dudley

Publisher: MAA

ISBN: 9780883853474

Category: Mathematics

Page: 141

View: 3754

"A Guide to Elementary Number Theory is a 140-page exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through traditional texts, some of which approach 500 pages in length. It will be especially useful to graduate students preparing for qualifying exams. Though Plato did not quite say, "He is unworthy of the name of man who does not know which integers are the sums of two squares," he came close. This guide can make everyone more worthy."--P. [4] of cover.

Elementary Theory of Numbers

Author: William J. LeVeque

Publisher: Courier Corporation

ISBN: 0486150763

Category: Mathematics

Page: 160

View: 4240

Superb introduction to Euclidean algorithm and its consequences, congruences, continued fractions, powers of an integer modulo m, Gaussian integers, Diophantine equations, more. Problems, with answers. Bibliography.

Elementary Number Theory, Group Theory and Ramanujan Graphs

Author: Giuliana Davidoff,Peter Sarnak,Alain Valette

Publisher: Cambridge University Press

ISBN: 9780521531436

Category: Mathematics

Page: 144

View: 2088

A self-contained treatment of expander graphs which are important in computer science, engineering and mathematics.

Elementary Number Theory and Its Applications

Author: Kenneth H. Rosen

Publisher: N.A

ISBN: 9780321717757

Category: Mathematics

Page: 752

View: 933

This text blends classical theory with modern applications and is notable for its outstanding exercise sets. A full range of exercises, from basic to challenging, helps students explore key concepts and push their understanding to new heights.

Elementary Number Theory

Author: Edmund Landau

Publisher: Taylor & Francis US

ISBN: 9780821820049

Category: Mathematics

Page: 256

View: 6981

This is a translation of Landau's famous Elementare Zahlentheorie with added exercises by Paul T. Bateman and Eugene E. Kohlbecker. This three-volume classic work is reprinted here as a single volume.